History
Dijkstra thought about the shortest path problem when working at the Mathematical Center in Amsterdam in 1956 as a programmer to demonstrate the capabilities of a new computer called ARMAC. His objective was to choose both a problem and a solution (that would be produced by computer) that non-computing people could understand. He designed the shortest path algorithm and later implemented it for ARMAC for a slightly simplified transportation map of 64 cities in the Netherlands (64, so that 6 bits would be sufficient to encode the city number). A year later, he came across another problem from hardware engineers working on the institute's next computer: minimize the amount of wire needed to connect the pins on the back panel of the machine. As a solution, he re-discovered the algorithm known as Prim's minimal spanning tree algorithm (known earlier to Jarník, and also rediscovered byAlgorithm
Let the node at which we are starting be called the initial node. Let the distance of node ''Y'' be the distance from the initial node to ''Y''. Dijkstra's algorithm will initially start with infinite distances and will try to improve them step by step. # Mark all nodes unvisited. Create aDescription
Suppose you would like to find the ''shortest path'' between two intersections on a city map: a ''starting point'' and a ''destination''. Dijkstra's algorithm initially marks the distance (from the starting point) to every other intersection on the map with ''infinity''. This is done not to imply that there is an infinite distance, but to note that those intersections have not been visited yet. Some variants of this method leave the intersections' distances ''unlabeled''. Now select the ''current intersection'' at each iteration. For the first iteration, the current intersection will be the starting point, and the distance to it (the intersection's label) will be ''zero''. For subsequent iterations (after the first), the current intersection will be a ''closest unvisited intersection'' to the starting point (this will be easy to find). From the current intersection, ''update'' the distance to every unvisited intersection that is directly connected to it. This is done by determining the ''sum'' of the distance between an unvisited intersection and the value of the current intersection and then relabeling the unvisited intersection with this value (the sum) if it is less than the unvisited intersection's current value. In effect, the intersection is relabeled if the path to it through the current intersection is shorter than the previously known paths. To facilitate shortest path identification, in pencil, mark the road with an arrow pointing to the relabeled intersection if you label/relabel it, and erase all others pointing to it. After you have updated the distances to each neighboring intersection, mark the current intersection as ''visited'' and select an unvisited intersection with minimal distance (from the starting point) – or the lowest label—as the current intersection. Intersections marked as visited are labeled with the shortest path from the starting point to it and will not be revisited or returned to. Continue this process of updating the neighboring intersections with the shortest distances, marking the current intersection as visited, and moving onto a closest unvisited intersection until you have marked the destination as visited. Once you have marked the destination as visited (as is the case with any visited intersection), you have determined the shortest path to it from the starting point and can ''trace your way back following the arrows in reverse''. In the algorithm's implementations, this is usually done (after the algorithm has reached the destination node) by following the nodes' parents from the destination node up to the starting node; that's why we also keep track of each node's parent. This algorithm makes no attempt of direct "exploration" towards the destination as one might expect. Rather, the sole consideration in determining the next "current" intersection is its distance from the starting point. This algorithm therefore expands outward from the starting point, interactively considering every node that is closer in terms of shortest path distance until it reaches the destination. When understood in this way, it is clear how the algorithm necessarily finds the shortest path. However, it may also reveal one of the algorithm's weaknesses: its relative slowness in some topologies.Pseudocode
In the following pseudocode algorithm, is an array that contains the current distances from the to other vertices, i.e. is the current distance from the source to the vertex . The array contains pointers to previous-hop nodes on the shortest path from source to the given vertex (equivalently, it is the ''next-hop'' on the path ''from'' the given vertex ''to'' the source). The code , searches for the vertex in the vertex set that has the least value. returns the length of the edge joining (i.e. the distance between) the two neighbor-nodes and . The variable on line 14 is the length of the path from the root node to the neighbor node if it were to go through . If this path is shorter than the current shortest path recorded for , that current path is replaced with this path. 1 function Dijkstra(''Graph'', ''source''): 2 3 for each vertex ''v'' in ''Graph.Vertices'': 4 dist 'v''← INFINITY 5 prev 'v''← UNDEFINED 6 add ''v'' to ''Q'' 7 dist 'source''← 0 8 9 while ''Q'' is not empty: 10 ''u'' ← vertex in ''Q'' with min dist 11 remove u from ''Q'' 12 13 for each neighbor ''v'' of ''u'' still in ''Q'': 14 ''alt'' ← dist 'u''+ Graph.Edges(''u'', ''v'') 15 if ''alt'' < dist 'v'' 16 dist 'v''← ''alt'' 17 prev 'v''← ''u'' 18 19 return dist[], prev[] If we are only interested in a shortest path between vertices and , we can terminate the search after line 10 if . Now we can read the shortest path from to by reverse iteration: 1 ''S'' ← empty sequence 2 ''u'' ← ''target'' 3 if prev 'u''is defined or ''u'' = ''source'': ''// Do something only if the vertex is reachable'' 4 while ''u'' is defined: ''// Construct the shortest path with a stack S'' 5 insert ''u'' at the beginning of ''S'' ''// Push the vertex onto the stack'' 6 ''u'' ← prev 'u'' ''// Traverse from target to source'' Now sequence is the list of vertices constituting one of the shortest paths from to , or the empty sequence if no path exists. A more general problem would be to find all the shortest paths between and (there might be several different ones of the same length). Then instead of storing only a single node in each entry of we would store all nodes satisfying the relaxation condition. For example, if both and connect to and both of them lie on different shortest paths through (because the edge cost is the same in both cases), then we would add both and to . When the algorithm completes, data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these shortest paths between two given nodes we would use a path finding algorithm on the new graph, such as depth-first search.Using a priority queue
A min-priority queue is an abstract data type that provides 3 basic operations: , and . As mentioned earlier, using such a data structure can lead to faster computing times than using a basic queue. Notably,if ''alt'' < dist 'v''/code> block, the becomes an operation if the node is not already in the queue.
Yet another alternative is to add nodes unconditionally to the priority queue and to instead check after extraction that no shorter connection was found yet. This can be done by additionally extracting the associated priority ''p''
from the queue and only processing further if ''p'' dist 'u''/code> inside the while ''Q'' is not empty
loop. Observe that cannot ever hold because of the update when updating the queue. See https://cs.stackexchange.com/questions/118388/dijkstra-without-decrease-key for discussion.
These alternatives can use entirely array-based priority queues without decrease-key functionality, which have been found to achieve even faster computing times in practice. However, the difference in performance was found to be narrower for denser graphs.
Proof of correctness
''Proof of Dijkstra's algorithm is constructed by induction on the number of visited nodes.''
''Invariant hypothesis'': For each visited node , is the shortest distance from to , and for each unvisited node , is the shortest distance from to when traveling via visited nodes only, or infinity if no such path exists. (Note: we do not assume is the actual shortest distance for unvisited nodes, while is the actual shortest distance)
The base case is when there is just one visited node, namely the initial node , in which case the hypothesis is trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
.
Next, assume the hypothesis for ''k-1'' visited nodes. Next, we choose to be the next visited node according to the algorithm. We claim that is the shortest distance from to .
To prove that claim, we will proceed with a proof by contradiction. If there were a shorter path, then there can be two cases, either the shortest path contains another unvisited node or not.
In the first case, let be the first unvisited node on the shortest path. By the induction hypothesis, the shortest path from to and through visited node only has cost and respectively. That means the cost of going from to through has the cost of at least + the minimal cost of going from to . As the edge costs are positive, the minimal cost of going from to is a positive number.
Also < because the algorithm picked instead of .
Now we arrived at a contradiction that < yet + a positive number < .
In the second case, let be the last but one node on the shortest path. That means . That is a contradiction because by the time is visited, it should have set to at most .
For all other visited nodes , the induction hypothesis told us is the shortest distance from already, and the algorithm step is not changing that.
After processing it will still be true that for each unvisited node , will be the shortest distance from to using visited nodes only, because if there were a shorter path that doesn't go by we would have found it previously, and if there were a shorter path using we would have updated it when processing .
After all nodes are visited, the shortest path from to any node consists only of visited nodes, therefore is the shortest distance.
Running time
Bounds of the running time of Dijkstra's algorithm on a graph with edges and vertices can be expressed as a function of the number of edges, denoted , and the number of vertices, denoted , using big-O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
. The complexity bound depends mainly on the data structure used to represent the set . In the following, upper bounds can be simplified because is for any graph, but that simplification disregards the fact that in some problems, other upper bounds on may hold.
For any data structure for the vertex set , the running time is in
:
where and are the complexities of the ''decrease-key'' and ''extract-minimum'' operations in , respectively.
The simplest version of Dijkstra's algorithm stores the vertex set as an linked list or array, and edges as an adjacency list
In graph theory and computer science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the set of neighbors of a particular vertex in the graph. This is ...
or matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. In this case, extract-minimum is simply a linear search through all vertices in , so the running time is .
For sparse graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction ...
s, that is, graphs with far fewer than edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a self-balancing binary search tree
In computer science, a self-balancing binary search tree (BST) is any node-based binary search tree that automatically keeps its height (maximal number of levels below the root) small in the face of arbitrary item insertions and deletions.Donal ...
, binary heap
A binary heap is a heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. The binary heap was introduced by J. W. J. Williams in 1964, as a data structure for heapsort.
A bin ...
, pairing heap
A pairing heap is a type of heap data structure with relatively simple implementation and excellent practical amortized performance, introduced by Michael Fredman, Robert Sedgewick, Daniel Sleator, and Robert Tarjan in 1986.
Pairing heaps are ...
, or Fibonacci heap
In computer science, a Fibonacci heap is a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binar ...
as a priority queue to implement extracting minimum efficiently. To perform decrease-key steps in a binary heap efficiently, it is necessary to use an auxiliary data structure that maps each vertex to its position in the heap, and to keep this structure up to date as the priority queue changes. With a self-balancing binary search tree or binary heap, the algorithm requires
:
time in the worst case (where denotes the binary logarithm ); for connected graphs this time bound can be simplified to . The Fibonacci heap
In computer science, a Fibonacci heap is a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binar ...
improves this to
:
When using binary heaps, the average case
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, b ...
time complexity is lower than the worst-case: assuming edge costs are drawn independently from a common probability distribution, the expected number of ''decrease-key'' operations is bounded by , giving a total running time of
:
Practical optimizations and infinite graphs
In common presentations of Dijkstra's algorithm, initially all nodes are entered into the priority queue. This is, however, not necessary: the algorithm can start with a priority queue that contains only one item, and insert new items as they are discovered (instead of doing a decrease-key, check whether the key is in the queue; if it is, decrease its key, otherwise insert it). This variant has the same worst-case bounds as the common variant, but maintains a smaller priority queue in practice, speeding up the queue operations. In a route-finding problem, Felner finds that the queue can be a factor 500–600 smaller, taking some 40% of the running time.
Moreover, not inserting all nodes in a graph makes it possible to extend the algorithm to find the shortest path from a single source to the closest of a set of target nodes on infinite graphs or those too large to represent in memory. The resulting algorithm is called ''uniform-cost search'' (UCS) in the artificial intelligence literature and can be expressed in pseudocode as
procedure uniform_cost_search(start) is
node ← start
frontier ← priority queue containing node only
expanded ← empty set
do
if frontier is empty then
return failure
node ← frontier.pop()
if node is a goal state then
return solution(node)
expanded.add(node)
for each of node's neighbors ''n'' do
if ''n'' is not in expanded and not in frontier then
frontier.add(''n'')
else if ''n'' is in frontier with higher cost
replace existing node with ''n''
The complexity of this algorithm can be expressed in an alternative way for very large graphs: when is the length of the shortest path from the start node to any node satisfying the "goal" predicate, each edge has cost at least , and the number of neighbors per node is bounded by , then the algorithm's worst-case time and space complexity are both in .
Further optimizations of Dijkstra's algorithm for the single-target case include bidirectional
Bidirectional may refer to:
* Bidirectional, a roadway that carries traffic moving in opposite directions
* Bi-directional vehicle, a tram or train or any other vehicle that can be controlled from either end and can move forward or backward with e ...
variants, goal-directed variants such as the A* algorithm (see ), graph pruning to determine which nodes are likely to form the middle segment of shortest paths (reach-based routing), and hierarchical decompositions of the input graph that reduce routing to connecting and to their respective " transit nodes" followed by shortest-path computation between these transit nodes using a "highway".
Combinations of such techniques may be needed for optimal practical performance on specific problems.
Specialized variants
When arc weights are small integers (bounded by a parameter ), specialized queues which take advantage of this fact can be used to speed up Dijkstra's algorithm. The first algorithm of this type was Dial's algorithm for graphs with positive integer edge weights, which uses a bucket queue
A bucket queue is a data structure that implements the priority queue abstract data type: it maintains a dynamic collection of elements with numerical priorities and allows quick access to the element with minimum (or maximum) priority. In the bu ...
to obtain a running time . The use of a Van Emde Boas tree
A van Emde Boas tree (), also known as a vEB tree or van Emde Boas priority queue, is a tree data structure which implements an associative array with -bit integer keys. It was invented by a team led by Dutch computer scientist Peter van Emde Boa ...
as the priority queue brings the complexity to . Another interesting variant based on a combination of a new radix heap
A radix heap is a data structure for realizing the operations of a monotone priority queue. A set of elements to which a key is assigned can then be managed. The run time of the operations depends on the difference between the largest and smalles ...
and the well-known Fibonacci heap runs in time . Finally, the best algorithms in this special case are as follows. The algorithm given by runs in time and the algorithm given by runs in time.
Related problems and algorithms
The functionality of Dijkstra's original algorithm can be extended with a variety of modifications. For example, sometimes it is desirable to present solutions which are less than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortest-path calculated. The secondary solutions are then ranked and presented after the first optimal solution.
Dijkstra's algorithm is usually the working principle behind link-state routing protocol
Link-state routing protocols are one of the two main classes of routing protocols used in packet switching networks for computer communications, the others being distance-vector routing protocols. Examples of link-state routing protocols includ ...
s, OSPF
Open Shortest Path First (OSPF) is a routing protocol for Internet Protocol (IP) networks. It uses a link state routing (LSR) algorithm and falls into the group of interior gateway protocols (IGPs), operating within a single autonomous syst ...
and IS-IS
Intermediate System to Intermediate System (IS-IS, also written ISIS) is a routing protocol designed to move information efficiently within a computer network, a group of physically connected computers or similar devices. It accomplishes this b ...
being the most common ones.
Unlike Dijkstra's algorithm, the Bellman–Ford algorithm
The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph.
It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it i ...
can be used on graphs with negative edge weights, as long as the graph contains no negative cycle reachable from the source vertex ''s''. The presence of such cycles means there is no shortest path, since the total weight becomes lower each time the cycle is traversed. (This statement assumes that a "path" is allowed to repeat vertices. In graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
that is normally not allowed. In theoretical computer science
computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumscribe the ...
it often is allowed.) It is possible to adapt Dijkstra's algorithm to handle negative weight edges by combining it with the Bellman-Ford algorithm (to remove negative edges and detect negative cycles); such an algorithm is called Johnson's algorithm
Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using ...
.
The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. This approach can be viewed from the perspective of linear programming: there is a natural linear program for computing shortest paths, and solutions to its dual linear program The dual of a given linear program (LP) is another LP that is derived from the original (the primal) LP in the following schematic way:
* Each variable in the primal LP becomes a constraint in the dual LP;
* Each constraint in the primal LP becomes ...
are feasible if and only if they form a consistent heuristic
In the study of path-finding problems in artificial intelligence, a heuristic function is said to be consistent, or monotone, if its estimate is always less than or equal to the estimated distance from any neighbouring vertex to the goal, plus the ...
(speaking roughly, since the sign conventions differ from place to place in the literature). This feasible dual / consistent heuristic defines a non-negative reduced cost
In linear programming, reduced cost, or opportunity cost, is the amount by which an objective function coefficient would have to improve (so increase for maximization problem, decrease for minimization problem) before it would be possible for a c ...
and A* is essentially running Dijkstra's algorithm with these reduced costs. If the dual satisfies the weaker condition of admissibility, then A* is instead more akin to the Bellman–Ford algorithm.
The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm
In computer science, Prim's algorithm (also known as Jarník's algorithm) is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every v ...
. Prim's purpose is to find a minimum spanning tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. T ...
that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual edges.
Breadth-first search
Breadth-first search (BFS) is an algorithm for searching a tree data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next de ...
can be viewed as a special-case of Dijkstra's algorithm on unweighted graphs, where the priority queue degenerates into a FIFO queue.
The fast marching method
The fast marching methodJ.A. Sethian. A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Natl. Acad. Sci., 93, 4, pp.1591--1595, 1996/ref> is a numerical method created by James Sethian for solving boundary value problems ...
can be viewed as a continuous version of Dijkstra's algorithm which computes the geodesic distance on a triangle mesh.
Dynamic programming perspective
From a dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. ...
point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.Online version of the paper with interactive computational modules.
/ref>
In fact, Dijkstra's explanation of the logic behind the algorithm, namely
is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.
Applications
Least-cost paths are calculated for instance to establish tracks of electricity lines or oil pipelines. The algorithm has also been used to calculate optimal long-distance footpaths in Ethiopia and contrast them with the situation on the ground.Nyssen, J., Tesfaalem Ghebreyohannes, Hailemariam Meaza, Dondeyne, S., 2020. Exploration of a medieval African map (Aksum, Ethiopia) – How do historical maps fit with topography? In: De Ryck, M., Nyssen, J., Van Acker, K., Van Roy, W., Liber Amicorum: Philippe De Maeyer In Kaart. Wachtebeke (Belgium): University Press: 165-178.
See also
* A* search algorithm
A* (pronounced "A-star") is a graph traversal and path search algorithm, which is used in many fields of computer science due to its completeness, optimality, and optimal efficiency. One major practical drawback is its O(b^d) space complexity, ...
* Bellman–Ford algorithm
The Bellman–Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph.
It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it i ...
* Euclidean shortest path
The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles.
Two d ...
* Floyd–Warshall algorithm
In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with p ...
* Johnson's algorithm
Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using ...
* Longest path problem
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called ''simple'' if it does not have any repeated vertices; the length of a path ma ...
* Parallel all-pairs shortest path algorithm
Notes
References
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External links
Oral history interview with Edsger W. Dijkstra
Charles Babbage Institute
The IT History Society (ITHS) is an organization that supports the history and scholarship of information technology by encouraging, fostering, and facilitating archival and historical research. Formerly known as the Charles Babbage Foundation, ...
, University of Minnesota, Minneapolis
Implementation of Dijkstra's algorithm using TDD
Robert Cecil Martin
Robert Cecil Martin (born 5 December 1952), colloquially called "Uncle Bob", is an American software engineer, instructor, and best-selling author. He is most recognized for developing many software design principles and for being a founder of t ...
, The Clean Code Blog
{{Optimization algorithms
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
1959 in computing
Graph algorithms
Search algorithms
Routing algorithms
Combinatorial optimization
Articles with example pseudocode
Dutch inventions
Graph distance